Theorem relcmpcmet 24835

Description: If 𝐷 is a metric space such that all the balls of some fixed size are relatively compact, then 𝐷 is complete. (Contributed by Mario Carneiro, 15-Oct-2015.)

Hypotheses

Ref	Expression
relcmpcmet.1	⊢ 𝐽 = (MetOpen‘𝐷)
relcmpcmet.2	⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
relcmpcmet.3	⊢ (𝜑 → 𝑅 ∈ ℝ+)
relcmpcmet.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)

Assertion

Ref	Expression
relcmpcmet	⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))

Distinct variable groups:   𝑥,𝐷   𝑥,𝐽   𝜑,𝑥   𝑥,𝑅   𝑥,𝑋

Proof of Theorem relcmpcmet

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relcmpcmet.2	. 2 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
2		metxmet 23840	. . . . . . 7 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
4	3	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝐷 ∈ (∞Met‘𝑋))
5		simpr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝑓 ∈ (CauFil‘𝐷))
6		relcmpcmet.3	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℝ+)
7	6	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝑅 ∈ ℝ+)
8		cfil3i 24786	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑓 ∈ (CauFil‘𝐷) ∧ 𝑅 ∈ ℝ+) → ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)
9	4, 5, 7, 8	syl3anc 1372	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)
10	3	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝐷 ∈ (∞Met‘𝑋))
11		relcmpcmet.1	. . . . . . . . 9 ⊢ 𝐽 = (MetOpen‘𝐷)
12	11	mopntopon 23945	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
13	10, 12	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝐽 ∈ (TopOn‘𝑋))
14		cfilfil 24784	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝑓 ∈ (Fil‘𝑋))
15	3, 14	sylan 581	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝑓 ∈ (Fil‘𝑋))
16	15	adantr 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑓 ∈ (Fil‘𝑋))
17		simprr 772	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)
18		topontop 22415	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
19	13, 18	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝐽 ∈ Top)
20		simprl 770	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑥 ∈ 𝑋)
21	6	rpxrd 13017	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℝ*)
22	21	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑅 ∈ ℝ*)
23		blssm 23924	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑥(ball‘𝐷)𝑅) ⊆ 𝑋)
24	10, 20, 22, 23	syl3anc 1372	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ⊆ 𝑋)
25		toponuni 22416	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
26	13, 25	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑋 = ∪ 𝐽)
27	24, 26	sseqtrd 4023	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ⊆ ∪ 𝐽)
28		eqid 2733	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
29	28	clsss3 22563	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (𝑥(ball‘𝐷)𝑅) ⊆ ∪ 𝐽) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ ∪ 𝐽)
30	19, 27, 29	syl2anc 585	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ ∪ 𝐽)
31	30, 26	sseqtrrd 4024	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋)
32	28	sscls 22560	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑥(ball‘𝐷)𝑅) ⊆ ∪ 𝐽) → (𝑥(ball‘𝐷)𝑅) ⊆ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))
33	19, 27, 32	syl2anc 585	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ⊆ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))
34		filss 23357	. . . . . . . 8 ⊢ ((𝑓 ∈ (Fil‘𝑋) ∧ ((𝑥(ball‘𝐷)𝑅) ∈ 𝑓 ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ⊆ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓)
35	16, 17, 31, 33, 34	syl13anc 1373	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓)
36		fclsrest 23528	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓) → ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) = ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
37	13, 16, 35, 36	syl3anc 1372	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) = ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
38		inss1 4229	. . . . . . 7 ⊢ ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ⊆ (𝐽 fClus 𝑓)
39		eqid 2733	. . . . . . . . 9 ⊢ dom dom 𝐷 = dom dom 𝐷
40	11, 39	cfilfcls 24791	. . . . . . . 8 ⊢ (𝑓 ∈ (CauFil‘𝐷) → (𝐽 fClus 𝑓) = (𝐽 fLim 𝑓))
41	40	ad2antlr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 fClus 𝑓) = (𝐽 fLim 𝑓))
42	38, 41	sseqtrid 4035	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ⊆ (𝐽 fLim 𝑓))
43	37, 42	eqsstrd 4021	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ⊆ (𝐽 fLim 𝑓))
44		relcmpcmet.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)
45	44	ad2ant2r 746	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)
46		filfbas 23352	. . . . . . . . . 10 ⊢ (𝑓 ∈ (Fil‘𝑋) → 𝑓 ∈ (fBas‘𝑋))
47	16, 46	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑓 ∈ (fBas‘𝑋))
48		fbncp 23343	. . . . . . . . 9 ⊢ ((𝑓 ∈ (fBas‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓) → ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓)
49	47, 35, 48	syl2anc 585	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓)
50		trfil3 23392	. . . . . . . . 9 ⊢ ((𝑓 ∈ (Fil‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋) → ((𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ↔ ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓))
51	16, 31, 50	syl2anc 585	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ↔ ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓))
52	49, 51	mpbird 257	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
53		resttopon 22665	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (TopOn‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
54	13, 31, 53	syl2anc 585	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (TopOn‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
55		toponuni 22416	. . . . . . . . 9 ⊢ ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (TopOn‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) = ∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
56	54, 55	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) = ∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
57	56	fveq2d 6896	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) = (Fil‘∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))))
58	52, 57	eleqtrd 2836	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))))
59		eqid 2733	. . . . . . 7 ⊢ ∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) = ∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))
60	59	fclscmpi 23533	. . . . . 6 ⊢ (((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp ∧ (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))) → ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ≠ ∅)
61	45, 58, 60	syl2anc 585	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ≠ ∅)
62		ssn0 4401	. . . . 5 ⊢ ((((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ⊆ (𝐽 fLim 𝑓) ∧ ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ≠ ∅) → (𝐽 fLim 𝑓) ≠ ∅)
63	43, 61, 62	syl2anc 585	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 fLim 𝑓) ≠ ∅)
64	9, 63	rexlimddv 3162	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → (𝐽 fLim 𝑓) ≠ ∅)
65	64	ralrimiva 3147	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅)
66	11	iscmet 24801	. 2 ⊢ (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))
67	1, 65, 66	sylanbrc 584	1 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ∀wral 3062  ∃wrex 3071   ∖ cdif 3946   ∩ cin 3948   ⊆ wss 3949  ∅c0 4323  ∪ cuni 4909  dom cdm 5677  ‘cfv 6544  (class class class)co 7409  ℝ^*cxr 11247  ℝ⁺crp 12974   ↾_t crest 17366  ∞Metcxmet 20929  Metcmet 20930  ballcbl 20931  fBascfbas 20932  MetOpencmopn 20934  Topctop 22395  TopOnctopon 22412  clsccl 22522  Compccmp 22890  Filcfil 23349   fLim cflim 23438   fClus cfcls 23440  CauFilccfil 24769  CMetccmet 24771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fi 9406  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-n0 12473  df-z 12559  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-ico 13330  df-rest 17368  df-topgen 17389  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-mopn 20940  df-fbas 20941  df-fg 20942  df-top 22396  df-topon 22413  df-bases 22449  df-cld 22523  df-ntr 22524  df-cls 22525  df-nei 22602  df-cmp 22891  df-fil 23350  df-flim 23443  df-fcls 23445  df-cfil 24772  df-cmet 24774

This theorem is referenced by:  cmpcmet  24836  cncmet  24839